The degree excess function $\epsilon(I,n)$ is the difference between the maximal generating degree $d(I^n)$ of the n-th power of a homogeneous ideal $I$ of a polynomial ring and $p(I)n$, where $p(I)$ is the leading coefficient of the asymptotically linear function $d(I^n)$. It is shown that any non-increasing numerical function can be realized as a degree excess function, and there is a monomial ideal I whose $\epsilon(I,n)$ has exactly a given number of local maxima. In the case of monomial ideals, an upper bound on $\epsilon(I,n)$ is provided. As an application, it is shown that in the worst case, the so-called stability index of the Castelnuovo-Mumford regularity of a monomial ideal $$ must be at least an exponential function of the number of variables.